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## Über dieses Buch

This book is intended for &tudents, research engineers, and mathematicians interested in applications or numerical analysis. Pure analysts will also find some new problems to tackle. Most of the material can be understood by a reader with a relatively modest knowledge of differential and inte­ gral equations and functional analysis. Readers interested in stochastic optimization will find a new theory of prac­ tical . importance. Readers interested in problems of static and quasi-static electrodynamics, wave scattering by small bodies of arbitrary shape, and corresponding applications in geophysics, optics, and radiophysics will find explicit analytical formulas for the scattering matrix, polarizability tensor, electrical capacitance of bodies of an arbitrary shape; numerical examples showing the practical utility of these formulas; two-sided variational estimates for the pol­ arizability tensor; and some open problems such as working out a standard program for calculating the capacitance and polarizability of bodies of arbitrary shape and numerical calculation of multiple integrals with weak singularities. Readers interested in nonlinear vibration theory will find a new method for qualitative study of stationary regimes in the general one-loop passive nonlinear network, including stabil­ ity in the large, convergence, and an iterative process for calculation the stationary regime. No assumptions concerning the smallness of the nonlinearity or the filter property of the linear one-port are made. New results in the theory of nonlinear operator equations form the basis for the study.

## Inhaltsverzeichnis

### Introduction

Abstract
There are many books and papers on integral equations. So the author should first explain why he has written a new book on the subject. Briefly, the explanation is as follows. Almost all the results presented in this book are new. Some new classes of integral equations are defined and investigated in this book. All the equations are closely connected with problems of physics and technology of great interest in applications. Some of the problems which have remained unsolved for years are solved in this book for the first time. Here we mention only three of them (Chapters 1–3);
A. G. Ramm

### Chapter I. Investigation of a New Class of Integral Equations and Applications to Estimation Problems (Filtering, Prediction, System Identification)

Abstract
Kolmogorov [1] initiated the study of filtering and extrapolation of stationary time series. These and other related problems were studied by N. Wiener in 1942 for stationary random processes and his results were published later in Wiener [1]. The basic integral equation of the theory of stochastic optimization for random processes is
$$Rh = \int_{{t - T}}^t {R(x,y)h(y)dy = f(x),\quad t - T \leqslant x \leqslant t}$$
(1.1)
where R(x,y) is a nonegative definite kernel, a correlation function, f(x) is a given function, and T > 0 is a given number. In Wiener [1] equation (1.1) was studied under the assumptions that R(x,y) = R(x-y) and T = +∞. We note that in applications T is the time of signal processing and the assumption about the kernel means that only stationary random processes were studied in Wiener [1]. Under these and some additional assumptions concerning the kernel R(x) a theory of the integral equation (1.1), now widely known as the Wiener-Hopf method, was given in Wiener-Hopf [1]. Their results were developed later in Krein [1], Gohberg-Krein [1], and Gohberg-Feldman [1].
A. G. Ramm

### Chapter II. Investigation of Integral Equations of the Static and Quasi-Static Fields and Applications to the Scattering from Small Bodies of Arbitrary Shape

Abstract
The calculation of static fields and some functionals of such fields, for example electrical capacitance or tensor polarizability, is of great interest in many applications. In particular, it is of basic interest for wave scattering by small bodies of arbitrary shape. Since the theory was initiated by Rayleigh [1] in 1871, very many papers have been published on this topic. Nevertheless the theory seemed incomplete in the following respect. Though wave scattering by a small body is a well understood process from the physical point of view there were no analytical formulas for the scattered field, scattering matrix, etc. In this chapter we obtain analytical formulas for the scattering matrix for the problems of scalar and vector wave scattering by a small body of arbitrary shape and by a system of such bodies. Analytical formulas for the calculation of the capacitance and polarizability of bodies of arbitrary shape with the needed accuracy are obtained. Two-sided variational estimates for the capacitance and polarizability are given. The formulas mentioned above are of immediate use in applications.
A. G. Ramm

### Chapter III. Investigation of a Class of Nonlinear Integral Equations and Applications to Nonlinear Network Theory

Abstract
Nonlinear oscillations in some networks can be described by the operator equation Au + Fu = J, where A is an unbounded linear operator on a Hilbert or Banach space, F is a nonlinear operator, and B = A + F is monotone. This is true for the general passive one-loop network, consisting of an e.m.f. E(t), and arbitrary linear passive stable one-port L, and a nonlinear one-port N with a monotone voltage-current characteristic i = Fu, where i is the current through N and u is the voltage on N. The theory presented below makes it possible to study nonlinear oscillations qualitatively, including questions of existence, uniqueness, stability in the large, convergence, and calculation of stationary regimes by means of an iterative process. Our assumptions concerning the network are more general than those usually adopted in the literature. No assumptions concerning the “smallness” of the nonlinearity or filter property of the linear one-port are made. Our results for nonlinear networks of the class defined above are final in the sense that if we omit the assumption concerning passivity of the network the results will not hold.
A. G. Ramm

### Chapter IV. Integral Equations Arising in the Open System Theory

Abstract
In quantum mechanics, in potential scattering theory, and in diffraction theory it is important to know the complex poles of Green’s functions. These poles determine energy losses in open systems, and are called resonances in quantum mechanics (see Baz et al. [1], Lifschitz [1]). Here we give a general method for numerical calculation of these complex poles. The method is described for quantum mechanics scattering problems and for diffraction problems.
A. G. Ramm

### Chapter V. Investigation of Some Integral Equations Arising in Antenna Synthesis

Abstract
Let A be a compact linear operator on a Hubert space $$H,\,N(A) = \left\{ 0 \right\},\,A{\phi_n} = {\lambda_n}{\phi_n},\,\left| {{\lambda_1}} \right| \geqslant \left| {{\lambda_2}} \right| \geqslant ...$$. We assume that the system $$\left\{ {{\phi_n}} \right\}$$ is an orthonormal basis of H.
A. G. Ramm

### Backmatter

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